Notes on Statistics: ISLR Lab Samples
- Chapter 3: Linear Regression
- Chapter 4: Logistic Regression, LDA, QDA, and KNN
- Chapter 5: Cross-Validation and the Bootstrap
- Chapter 6
- Subset Selection Methods: Best Subset Selection
- Subset Selection Methods: Forward and Backward Stepwise Selection
- Subset Selection Methods: Choosing Among Models
- Ridge Regression and the Lasso
- Ridge Regression and the Lasso: Ridge Regression
- Ridge Regression and the Lasso: The Lasso
- PCR and PLS Regression : Principal Components Regression
- PCR and PLS Regression : Partial Least Squares
- Chapter 7: Non-linear Modeling
- Chapter 8: Decision Trees
- Chapter 9: Support Vector Machines
- Chapter 10
Samples from http://www-bcf.usc.edu/~gareth/ISL/code.html Python version https://github.com/sujitpal/statlearning-notebooks
library(MASS)
library(ISLR)
library(car)
library(class)Chapter 3: Linear Regression
Simple Linear Regression
names (Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"attach(Boston)
(lm.fit = lm (medv~lstat, data=Boston))##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## Coefficients:
## (Intercept) lstat
## 34.55 -0.95summary(lm.fit)##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.168 -3.990 -1.318 2.034 24.500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.55384 0.56263 61.41 <2e-16 ***
## lstat -0.95005 0.03873 -24.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.216 on 504 degrees of freedom
## Multiple R-squared: 0.5441, Adjusted R-squared: 0.5432
## F-statistic: 601.6 on 1 and 504 DF, p-value: < 2.2e-16names(lm.fit)## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"coef(lm.fit)## (Intercept) lstat
## 34.5538409 -0.9500494confint(lm.fit)## 2.5 % 97.5 %
## (Intercept) 33.448457 35.6592247
## lstat -1.026148 -0.8739505predict(lm.fit,data.frame(lstat=(c(5,10,15))), interval="confidence")## fit lwr upr
## 1 29.80359 29.00741 30.59978
## 2 25.05335 24.47413 25.63256
## 3 20.30310 19.73159 20.87461predict(lm.fit,data.frame(lstat=(c(5,10,15))), interval="prediction")## fit lwr upr
## 1 29.80359 17.565675 42.04151
## 2 25.05335 12.827626 37.27907
## 3 20.30310 8.077742 32.52846plot(lstat,medv)
abline(lm.fit)
abline(lm.fit,lwd=3)
abline(lm.fit,lwd=3,col="red")
plot(lstat,medv,col="red")
plot(lstat,medv,pch=20)
plot(lstat,medv,pch="+")
plot(1:20,1:20,pch=1:20)
par(mfrow=c(2,2))
plot(lm.fit)
plot(predict(lm.fit), residuals(lm.fit))
plot(predict(lm.fit), rstudent(lm.fit))
plot(hatvalues(lm.fit))
which.max(hatvalues(lm.fit))## 375
## 375
Multiple Linear Regression
lm.fit=lm(medv~lstat+age,data=Boston)
summary(lm.fit)##
## Call:
## lm(formula = medv ~ lstat + age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.981 -3.978 -1.283 1.968 23.158
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.22276 0.73085 45.458 < 2e-16 ***
## lstat -1.03207 0.04819 -21.416 < 2e-16 ***
## age 0.03454 0.01223 2.826 0.00491 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.173 on 503 degrees of freedom
## Multiple R-squared: 0.5513, Adjusted R-squared: 0.5495
## F-statistic: 309 on 2 and 503 DF, p-value: < 2.2e-16lm.fit=lm(medv~.,data=Boston)
summary(lm.fit)##
## Call:
## lm(formula = medv ~ ., data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.595 -2.730 -0.518 1.777 26.199
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.646e+01 5.103e+00 7.144 3.28e-12 ***
## crim -1.080e-01 3.286e-02 -3.287 0.001087 **
## zn 4.642e-02 1.373e-02 3.382 0.000778 ***
## indus 2.056e-02 6.150e-02 0.334 0.738288
## chas 2.687e+00 8.616e-01 3.118 0.001925 **
## nox -1.777e+01 3.820e+00 -4.651 4.25e-06 ***
## rm 3.810e+00 4.179e-01 9.116 < 2e-16 ***
## age 6.922e-04 1.321e-02 0.052 0.958229
## dis -1.476e+00 1.995e-01 -7.398 6.01e-13 ***
## rad 3.060e-01 6.635e-02 4.613 5.07e-06 ***
## tax -1.233e-02 3.760e-03 -3.280 0.001112 **
## ptratio -9.527e-01 1.308e-01 -7.283 1.31e-12 ***
## black 9.312e-03 2.686e-03 3.467 0.000573 ***
## lstat -5.248e-01 5.072e-02 -10.347 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.745 on 492 degrees of freedom
## Multiple R-squared: 0.7406, Adjusted R-squared: 0.7338
## F-statistic: 108.1 on 13 and 492 DF, p-value: < 2.2e-16car::vif(lm.fit)## crim zn indus chas nox rm age dis
## 1.792192 2.298758 3.991596 1.073995 4.393720 1.933744 3.100826 3.955945
## rad tax ptratio black lstat
## 7.484496 9.008554 1.799084 1.348521 2.941491lm.fit1=lm(medv~.-age,data=Boston)
summary(lm.fit1)##
## Call:
## lm(formula = medv ~ . - age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.6054 -2.7313 -0.5188 1.7601 26.2243
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.436927 5.080119 7.172 2.72e-12 ***
## crim -0.108006 0.032832 -3.290 0.001075 **
## zn 0.046334 0.013613 3.404 0.000719 ***
## indus 0.020562 0.061433 0.335 0.737989
## chas 2.689026 0.859598 3.128 0.001863 **
## nox -17.713540 3.679308 -4.814 1.97e-06 ***
## rm 3.814394 0.408480 9.338 < 2e-16 ***
## dis -1.478612 0.190611 -7.757 5.03e-14 ***
## rad 0.305786 0.066089 4.627 4.75e-06 ***
## tax -0.012329 0.003755 -3.283 0.001099 **
## ptratio -0.952211 0.130294 -7.308 1.10e-12 ***
## black 0.009321 0.002678 3.481 0.000544 ***
## lstat -0.523852 0.047625 -10.999 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.74 on 493 degrees of freedom
## Multiple R-squared: 0.7406, Adjusted R-squared: 0.7343
## F-statistic: 117.3 on 12 and 493 DF, p-value: < 2.2e-16lm.fit1=update(lm.fit, ~.-age)Interaction Terms
summary(lm(medv~lstat*age,data=Boston))##
## Call:
## lm(formula = medv ~ lstat * age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.806 -4.045 -1.333 2.085 27.552
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.0885359 1.4698355 24.553 < 2e-16 ***
## lstat -1.3921168 0.1674555 -8.313 8.78e-16 ***
## age -0.0007209 0.0198792 -0.036 0.9711
## lstat:age 0.0041560 0.0018518 2.244 0.0252 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.149 on 502 degrees of freedom
## Multiple R-squared: 0.5557, Adjusted R-squared: 0.5531
## F-statistic: 209.3 on 3 and 502 DF, p-value: < 2.2e-16Non-linear Transformations of the Predictors
lm.fit2=lm(medv~lstat+I(lstat^2))
summary(lm.fit2)##
## Call:
## lm(formula = medv ~ lstat + I(lstat^2))
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.2834 -3.8313 -0.5295 2.3095 25.4148
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 42.862007 0.872084 49.15 <2e-16 ***
## lstat -2.332821 0.123803 -18.84 <2e-16 ***
## I(lstat^2) 0.043547 0.003745 11.63 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.524 on 503 degrees of freedom
## Multiple R-squared: 0.6407, Adjusted R-squared: 0.6393
## F-statistic: 448.5 on 2 and 503 DF, p-value: < 2.2e-16lm.fit=lm(medv~lstat)
anova(lm.fit,lm.fit2)## Analysis of Variance Table
##
## Model 1: medv ~ lstat
## Model 2: medv ~ lstat + I(lstat^2)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 504 19472
## 2 503 15347 1 4125.1 135.2 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1par(mfrow=c(2,2))
plot(lm.fit2)
lm.fit5=lm(medv~poly(lstat,5))
summary(lm.fit5)##
## Call:
## lm(formula = medv ~ poly(lstat, 5))
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.5433 -3.1039 -0.7052 2.0844 27.1153
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.5328 0.2318 97.197 < 2e-16 ***
## poly(lstat, 5)1 -152.4595 5.2148 -29.236 < 2e-16 ***
## poly(lstat, 5)2 64.2272 5.2148 12.316 < 2e-16 ***
## poly(lstat, 5)3 -27.0511 5.2148 -5.187 3.10e-07 ***
## poly(lstat, 5)4 25.4517 5.2148 4.881 1.42e-06 ***
## poly(lstat, 5)5 -19.2524 5.2148 -3.692 0.000247 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.215 on 500 degrees of freedom
## Multiple R-squared: 0.6817, Adjusted R-squared: 0.6785
## F-statistic: 214.2 on 5 and 500 DF, p-value: < 2.2e-16summary(lm(medv~log(rm),data=Boston))##
## Call:
## lm(formula = medv ~ log(rm), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.487 -2.875 -0.104 2.837 39.816
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -76.488 5.028 -15.21 <2e-16 ***
## log(rm) 54.055 2.739 19.73 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.915 on 504 degrees of freedom
## Multiple R-squared: 0.4358, Adjusted R-squared: 0.4347
## F-statistic: 389.3 on 1 and 504 DF, p-value: < 2.2e-16Qualitative Predictors
names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"lm.fit=lm(Sales~.+Income:Advertising+Price:Age,data=Carseats)
summary(lm.fit)##
## Call:
## lm(formula = Sales ~ . + Income:Advertising + Price:Age, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.9208 -0.7503 0.0177 0.6754 3.3413
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.5755654 1.0087470 6.519 2.22e-10 ***
## CompPrice 0.0929371 0.0041183 22.567 < 2e-16 ***
## Income 0.0108940 0.0026044 4.183 3.57e-05 ***
## Advertising 0.0702462 0.0226091 3.107 0.002030 **
## Population 0.0001592 0.0003679 0.433 0.665330
## Price -0.1008064 0.0074399 -13.549 < 2e-16 ***
## ShelveLocGood 4.8486762 0.1528378 31.724 < 2e-16 ***
## ShelveLocMedium 1.9532620 0.1257682 15.531 < 2e-16 ***
## Age -0.0579466 0.0159506 -3.633 0.000318 ***
## Education -0.0208525 0.0196131 -1.063 0.288361
## UrbanYes 0.1401597 0.1124019 1.247 0.213171
## USYes -0.1575571 0.1489234 -1.058 0.290729
## Income:Advertising 0.0007510 0.0002784 2.698 0.007290 **
## Price:Age 0.0001068 0.0001333 0.801 0.423812
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.011 on 386 degrees of freedom
## Multiple R-squared: 0.8761, Adjusted R-squared: 0.8719
## F-statistic: 210 on 13 and 386 DF, p-value: < 2.2e-16attach(Carseats)
contrasts(ShelveLoc)## Good Medium
## Bad 0 0
## Good 1 0
## Medium 0 1Chapter 4: Logistic Regression, LDA, QDA, and KNN
The Stock Market Data
names(Smarket)## [1] "Year" "Lag1" "Lag2" "Lag3" "Lag4" "Lag5"
## [7] "Volume" "Today" "Direction"dim(Smarket)## [1] 1250 9summary(Smarket)## Year Lag1 Lag2
## Min. :2001 Min. :-4.922000 Min. :-4.922000
## 1st Qu.:2002 1st Qu.:-0.639500 1st Qu.:-0.639500
## Median :2003 Median : 0.039000 Median : 0.039000
## Mean :2003 Mean : 0.003834 Mean : 0.003919
## 3rd Qu.:2004 3rd Qu.: 0.596750 3rd Qu.: 0.596750
## Max. :2005 Max. : 5.733000 Max. : 5.733000
## Lag3 Lag4 Lag5
## Min. :-4.922000 Min. :-4.922000 Min. :-4.92200
## 1st Qu.:-0.640000 1st Qu.:-0.640000 1st Qu.:-0.64000
## Median : 0.038500 Median : 0.038500 Median : 0.03850
## Mean : 0.001716 Mean : 0.001636 Mean : 0.00561
## 3rd Qu.: 0.596750 3rd Qu.: 0.596750 3rd Qu.: 0.59700
## Max. : 5.733000 Max. : 5.733000 Max. : 5.73300
## Volume Today Direction
## Min. :0.3561 Min. :-4.922000 Down:602
## 1st Qu.:1.2574 1st Qu.:-0.639500 Up :648
## Median :1.4229 Median : 0.038500
## Mean :1.4783 Mean : 0.003138
## 3rd Qu.:1.6417 3rd Qu.: 0.596750
## Max. :3.1525 Max. : 5.733000pairs(Smarket)
#cor(Smarket)
cor(Smarket[,-9])## Year Lag1 Lag2 Lag3 Lag4
## Year 1.00000000 0.029699649 0.030596422 0.033194581 0.035688718
## Lag1 0.02969965 1.000000000 -0.026294328 -0.010803402 -0.002985911
## Lag2 0.03059642 -0.026294328 1.000000000 -0.025896670 -0.010853533
## Lag3 0.03319458 -0.010803402 -0.025896670 1.000000000 -0.024051036
## Lag4 0.03568872 -0.002985911 -0.010853533 -0.024051036 1.000000000
## Lag5 0.02978799 -0.005674606 -0.003557949 -0.018808338 -0.027083641
## Volume 0.53900647 0.040909908 -0.043383215 -0.041823686 -0.048414246
## Today 0.03009523 -0.026155045 -0.010250033 -0.002447647 -0.006899527
## Lag5 Volume Today
## Year 0.029787995 0.53900647 0.030095229
## Lag1 -0.005674606 0.04090991 -0.026155045
## Lag2 -0.003557949 -0.04338321 -0.010250033
## Lag3 -0.018808338 -0.04182369 -0.002447647
## Lag4 -0.027083641 -0.04841425 -0.006899527
## Lag5 1.000000000 -0.02200231 -0.034860083
## Volume -0.022002315 1.00000000 0.014591823
## Today -0.034860083 0.01459182 1.000000000attach(Smarket)
plot(Volume)
Logistic Regression
glm.fit=glm(Direction~Lag1+Lag2+Lag3+Lag4+Lag5+Volume,data=Smarket,family=binomial)
summary(glm.fit)##
## Call:
## glm(formula = Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 +
## Volume, family = binomial, data = Smarket)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.446 -1.203 1.065 1.145 1.326
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.126000 0.240736 -0.523 0.601
## Lag1 -0.073074 0.050167 -1.457 0.145
## Lag2 -0.042301 0.050086 -0.845 0.398
## Lag3 0.011085 0.049939 0.222 0.824
## Lag4 0.009359 0.049974 0.187 0.851
## Lag5 0.010313 0.049511 0.208 0.835
## Volume 0.135441 0.158360 0.855 0.392
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1731.2 on 1249 degrees of freedom
## Residual deviance: 1727.6 on 1243 degrees of freedom
## AIC: 1741.6
##
## Number of Fisher Scoring iterations: 3coef(glm.fit)## (Intercept) Lag1 Lag2 Lag3 Lag4
## -0.126000257 -0.073073746 -0.042301344 0.011085108 0.009358938
## Lag5 Volume
## 0.010313068 0.135440659summary(glm.fit)$coef## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.126000257 0.24073574 -0.5233966 0.6006983
## Lag1 -0.073073746 0.05016739 -1.4565986 0.1452272
## Lag2 -0.042301344 0.05008605 -0.8445733 0.3983491
## Lag3 0.011085108 0.04993854 0.2219750 0.8243333
## Lag4 0.009358938 0.04997413 0.1872757 0.8514445
## Lag5 0.010313068 0.04951146 0.2082966 0.8349974
## Volume 0.135440659 0.15835970 0.8552723 0.3924004summary(glm.fit)$coef[,4]## (Intercept) Lag1 Lag2 Lag3 Lag4 Lag5
## 0.6006983 0.1452272 0.3983491 0.8243333 0.8514445 0.8349974
## Volume
## 0.3924004glm.probs=predict(glm.fit,type="response")
glm.probs[1:10]## 1 2 3 4 5 6 7
## 0.5070841 0.4814679 0.4811388 0.5152224 0.5107812 0.5069565 0.4926509
## 8 9 10
## 0.5092292 0.5176135 0.4888378contrasts(Direction)## Up
## Down 0
## Up 1glm.pred=rep("Down",1250)
glm.pred[glm.probs>.5]="Up"
table(glm.pred,Direction)## Direction
## glm.pred Down Up
## Down 145 141
## Up 457 507(507+145)/1250## [1] 0.5216mean(glm.pred==Direction)## [1] 0.5216train=(Year<2005)
Smarket.2005=Smarket[!train,]
dim(Smarket.2005)## [1] 252 9Direction.2005=Direction[!train]
glm.fit=glm(Direction~Lag1+Lag2+Lag3+Lag4+Lag5+Volume,data=Smarket,family=binomial,subset=train)
glm.probs=predict(glm.fit,Smarket.2005,type="response")
glm.pred=rep("Down",252)
glm.pred[glm.probs>.5]="Up"
table(glm.pred,Direction.2005)## Direction.2005
## glm.pred Down Up
## Down 77 97
## Up 34 44mean(glm.pred==Direction.2005)## [1] 0.4801587mean(glm.pred!=Direction.2005)## [1] 0.5198413glm.fit=glm(Direction~Lag1+Lag2,data=Smarket,family=binomial,subset=train)
glm.probs=predict(glm.fit,Smarket.2005,type="response")
glm.pred=rep("Down",252)
glm.pred[glm.probs>.5]="Up"
table(glm.pred,Direction.2005)## Direction.2005
## glm.pred Down Up
## Down 35 35
## Up 76 106mean(glm.pred==Direction.2005)## [1] 0.5595238106/(106+76)## [1] 0.5824176predict(glm.fit,newdata=data.frame(Lag1=c(1.2,1.5),Lag2=c(1.1,-0.8)),type="response")## 1 2
## 0.4791462 0.4960939Linear Discriminant Analysis
lda.fit=lda(Direction~Lag1+Lag2,data=Smarket,subset=train) #MASS
lda.fit## Call:
## lda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)
##
## Prior probabilities of groups:
## Down Up
## 0.491984 0.508016
##
## Group means:
## Lag1 Lag2
## Down 0.04279022 0.03389409
## Up -0.03954635 -0.03132544
##
## Coefficients of linear discriminants:
## LD1
## Lag1 -0.6420190
## Lag2 -0.5135293plot(lda.fit)
lda.pred=predict(lda.fit, Smarket.2005)
names(lda.pred)## [1] "class" "posterior" "x"lda.class=lda.pred$class
table(lda.class,Direction.2005)## Direction.2005
## lda.class Down Up
## Down 35 35
## Up 76 106mean(lda.class==Direction.2005)## [1] 0.5595238sum(lda.pred$posterior[,1]>=.5)## [1] 70sum(lda.pred$posterior[,1]<.5)## [1] 182lda.pred$posterior[1:20,1]## 999 1000 1001 1002 1003 1004 1005
## 0.4901792 0.4792185 0.4668185 0.4740011 0.4927877 0.4938562 0.4951016
## 1006 1007 1008 1009 1010 1011 1012
## 0.4872861 0.4907013 0.4844026 0.4906963 0.5119988 0.4895152 0.4706761
## 1013 1014 1015 1016 1017 1018
## 0.4744593 0.4799583 0.4935775 0.5030894 0.4978806 0.4886331lda.class[1:20]## [1] Up Up Up Up Up Up Up Up Up Up Up Down Up Up
## [15] Up Up Up Down Up Up
## Levels: Down Upsum(lda.pred$posterior[,1]>.9)## [1] 0Quadratic Discriminant Analysis
qda.fit=qda(Direction~Lag1+Lag2,data=Smarket,subset=train)
qda.fit## Call:
## qda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)
##
## Prior probabilities of groups:
## Down Up
## 0.491984 0.508016
##
## Group means:
## Lag1 Lag2
## Down 0.04279022 0.03389409
## Up -0.03954635 -0.03132544qda.class=predict(qda.fit,Smarket.2005)$class
table(qda.class,Direction.2005)## Direction.2005
## qda.class Down Up
## Down 30 20
## Up 81 121mean(qda.class==Direction.2005)## [1] 0.5992063K-Nearest Neighbors
#class
train.X=cbind(Lag1,Lag2)[train,]
test.X=cbind(Lag1,Lag2)[!train,]
train.Direction=Direction[train]
set.seed(1)
knn.pred=knn(train.X,test.X,train.Direction,k=1)
table(knn.pred,Direction.2005)## Direction.2005
## knn.pred Down Up
## Down 43 58
## Up 68 83(83+43)/252## [1] 0.5knn.pred=knn(train.X,test.X,train.Direction,k=3)
table(knn.pred,Direction.2005)## Direction.2005
## knn.pred Down Up
## Down 48 54
## Up 63 87mean(knn.pred==Direction.2005)## [1] 0.5357143An Application to Caravan Insurance Data
dim(Caravan)## [1] 5822 86attach(Caravan)
summary(Purchase)## No Yes
## 5474 348348/5822## [1] 0.05977327standardized.X=scale(Caravan[,-86])
var(Caravan[,1])## [1] 165.0378var(Caravan[,2])## [1] 0.1647078var(standardized.X[,1])## [1] 1var(standardized.X[,2])## [1] 1test=1:1000
train.X=standardized.X[-test,]
test.X=standardized.X[test,]
train.Y=Purchase[-test]
test.Y=Purchase[test]
set.seed(1)
knn.pred=knn(train.X,test.X,train.Y,k=1)
mean(test.Y!=knn.pred)## [1] 0.118mean(test.Y!="No")## [1] 0.059table(knn.pred,test.Y)## test.Y
## knn.pred No Yes
## No 873 50
## Yes 68 99/(68+9)## [1] 0.1168831knn.pred=knn(train.X,test.X,train.Y,k=3)
table(knn.pred,test.Y)## test.Y
## knn.pred No Yes
## No 920 54
## Yes 21 55/26## [1] 0.1923077knn.pred=knn(train.X,test.X,train.Y,k=5)
table(knn.pred,test.Y)## test.Y
## knn.pred No Yes
## No 930 55
## Yes 11 44/15## [1] 0.2666667glm.fit=glm(Purchase~.,data=Caravan,family=binomial,subset=-test)## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurredglm.probs=predict(glm.fit,Caravan[test,],type="response")
glm.pred=rep("No",1000)
glm.pred[glm.probs>.5]="Yes"
table(glm.pred,test.Y)## test.Y
## glm.pred No Yes
## No 934 59
## Yes 7 0glm.pred=rep("No",1000)
glm.pred[glm.probs>.25]="Yes"
table(glm.pred,test.Y)## test.Y
## glm.pred No Yes
## No 919 48
## Yes 22 1111/(22+11)## [1] 0.3333333Chapter 5: Cross-Validation and the Bootstrap
The Validation Set Approach
set.seed(1)
train=sample(392,196)
lm.fit=lm(mpg~horsepower,data=Auto,subset=train)
attach(Auto)
mean((mpg-predict(lm.fit,Auto))[-train]^2)## [1] 26.14142lm.fit2=lm(mpg~poly(horsepower,2),data=Auto,subset=train)
mean((mpg-predict(lm.fit2,Auto))[-train]^2)## [1] 19.82259lm.fit3=lm(mpg~poly(horsepower,3),data=Auto,subset=train)
mean((mpg-predict(lm.fit3,Auto))[-train]^2)## [1] 19.78252set.seed(2)
train=sample(392,196)
lm.fit=lm(mpg~horsepower,subset=train)
mean((mpg-predict(lm.fit,Auto))[-train]^2)## [1] 23.29559lm.fit2=lm(mpg~poly(horsepower,2),data=Auto,subset=train)
mean((mpg-predict(lm.fit2,Auto))[-train]^2)## [1] 18.90124lm.fit3=lm(mpg~poly(horsepower,3),data=Auto,subset=train)
mean((mpg-predict(lm.fit3,Auto))[-train]^2)## [1] 19.2574Leave-One-Out Cross-Validation
glm.fit=glm(mpg~horsepower,data=Auto)
coef(glm.fit)## (Intercept) horsepower
## 39.9358610 -0.1578447lm.fit=lm(mpg~horsepower,data=Auto)
coef(lm.fit)## (Intercept) horsepower
## 39.9358610 -0.1578447library(boot)##
## Attaching package: 'boot'
##
## The following object is masked from 'package:car':
##
## logitglm.fit=glm(mpg~horsepower,data=Auto)
cv.err=cv.glm(Auto,glm.fit)
cv.err$delta## [1] 24.23151 24.23114cv.error=rep(0,5)
for (i in 1:5){
glm.fit=glm(mpg~poly(horsepower,i),data=Auto)
cv.error[i]=cv.glm(Auto,glm.fit)$delta[1]
}
cv.error## [1] 24.23151 19.24821 19.33498 19.42443 19.03321k-Fold Cross-Validation
set.seed(17)
cv.error.10=rep(0,10)
for (i in 1:10){
glm.fit=glm(mpg~poly(horsepower,i),data=Auto)
cv.error.10[i]=cv.glm(Auto,glm.fit,K=10)$delta[1]
}
cv.error.10## [1] 24.20520 19.18924 19.30662 19.33799 18.87911 19.02103 18.89609
## [8] 19.71201 18.95140 19.50196The Bootstrap
alpha.fn=function(data,index){
X=data$X[index]
Y=data$Y[index]
return((var(Y)-cov(X,Y))/(var(X)+var(Y)-2*cov(X,Y)))
}
alpha.fn(Portfolio,1:100)## [1] 0.5758321set.seed(1)
alpha.fn(Portfolio,sample(100,100,replace=T))## [1] 0.5963833boot(Portfolio,alpha.fn,R=1000)##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = Portfolio, statistic = alpha.fn, R = 1000)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 0.5758321 -7.315422e-05 0.08861826Estimating the Accuracy of a Linear Regression Model
boot.fn=function(data,index)
return(coef(lm(mpg~horsepower,data=data,subset=index)))
boot.fn(Auto,1:392)## (Intercept) horsepower
## 39.9358610 -0.1578447set.seed(1)
boot.fn(Auto,sample(392,392,replace=T))## (Intercept) horsepower
## 38.7387134 -0.1481952boot.fn(Auto,sample(392,392,replace=T))## (Intercept) horsepower
## 40.0383086 -0.1596104boot(Auto,boot.fn,1000)##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = Auto, statistic = boot.fn, R = 1000)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 39.9358610 0.02972191 0.860007896
## t2* -0.1578447 -0.00030823 0.007404467summary(lm(mpg~horsepower,data=Auto))$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.9358610 0.717498656 55.65984 1.220362e-187
## horsepower -0.1578447 0.006445501 -24.48914 7.031989e-81boot.fn=function(data,index)
coefficients(lm(mpg~horsepower+I(horsepower^2),data=data,subset=index))
set.seed(1)
boot(Auto,boot.fn,1000)##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = Auto, statistic = boot.fn, R = 1000)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 56.900099702 6.098115e-03 2.0944855842
## t2* -0.466189630 -1.777108e-04 0.0334123802
## t3* 0.001230536 1.324315e-06 0.0001208339summary(lm(mpg~horsepower+I(horsepower^2),data=Auto))$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 56.900099702 1.8004268063 31.60367 1.740911e-109
## horsepower -0.466189630 0.0311246171 -14.97816 2.289429e-40
## I(horsepower^2) 0.001230536 0.0001220759 10.08009 2.196340e-21Chapter 6
Subset Selection Methods: Best Subset Selection
names(Hitters)## [1] "AtBat" "Hits" "HmRun" "Runs" "RBI"
## [6] "Walks" "Years" "CAtBat" "CHits" "CHmRun"
## [11] "CRuns" "CRBI" "CWalks" "League" "Division"
## [16] "PutOuts" "Assists" "Errors" "Salary" "NewLeague"dim(Hitters)## [1] 322 20sum(is.na(Hitters$Salary))## [1] 59Hitters=na.omit(Hitters)
dim(Hitters)## [1] 263 20sum(is.na(Hitters))## [1] 0library(leaps)
regfit.full=regsubsets(Salary~.,Hitters)
summary(regfit.full)## Subset selection object
## Call: regsubsets.formula(Salary ~ ., Hitters)
## 19 Variables (and intercept)
## Forced in Forced out
## AtBat FALSE FALSE
## Hits FALSE FALSE
## HmRun FALSE FALSE
## Runs FALSE FALSE
## RBI FALSE FALSE
## Walks FALSE FALSE
## Years FALSE FALSE
## CAtBat FALSE FALSE
## CHits FALSE FALSE
## CHmRun FALSE FALSE
## CRuns FALSE FALSE
## CRBI FALSE FALSE
## CWalks FALSE FALSE
## LeagueN FALSE FALSE
## DivisionW FALSE FALSE
## PutOuts FALSE FALSE
## Assists FALSE FALSE
## Errors FALSE FALSE
## NewLeagueN FALSE FALSE
## 1 subsets of each size up to 8
## Selection Algorithm: exhaustive
## AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits CHmRun CRuns
## 1 ( 1 ) " " " " " " " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 5 ( 1 ) "*" "*" " " " " " " " " " " " " " " " " " "
## 6 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " " "
## 7 ( 1 ) " " "*" " " " " " " "*" " " "*" "*" "*" " "
## 8 ( 1 ) "*" "*" " " " " " " "*" " " " " " " "*" "*"
## CRBI CWalks LeagueN DivisionW PutOuts Assists Errors NewLeagueN
## 1 ( 1 ) "*" " " " " " " " " " " " " " "
## 2 ( 1 ) "*" " " " " " " " " " " " " " "
## 3 ( 1 ) "*" " " " " " " "*" " " " " " "
## 4 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 5 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 6 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 7 ( 1 ) " " " " " " "*" "*" " " " " " "
## 8 ( 1 ) " " "*" " " "*" "*" " " " " " "regfit.full=regsubsets(Salary~.,data=Hitters,nvmax=19)
reg.summary=summary(regfit.full)
names(reg.summary)## [1] "which" "rsq" "rss" "adjr2" "cp" "bic" "outmat" "obj"reg.summary$rsq## [1] 0.3214501 0.4252237 0.4514294 0.4754067 0.4908036 0.5087146 0.5141227
## [8] 0.5285569 0.5346124 0.5404950 0.5426153 0.5436302 0.5444570 0.5452164
## [15] 0.5454692 0.5457656 0.5459518 0.5460945 0.5461159par(mfrow=c(2,2))
plot(reg.summary$rss,xlab="Number of Variables",ylab="RSS",type="l")
plot(reg.summary$adjr2,xlab="Number of Variables",ylab="Adjusted RSq",type="l")
which.max(reg.summary$adjr2)## [1] 11points(11,reg.summary$adjr2[11], col="red",cex=2,pch=20)
plot(reg.summary$cp,xlab="Number of Variables",ylab="Cp",type='l')
which.min(reg.summary$cp)## [1] 10points(10,reg.summary$cp[10],col="red",cex=2,pch=20)
which.min(reg.summary$bic)## [1] 6plot(reg.summary$bic,xlab="Number of Variables",ylab="BIC",type='l')
points(6,reg.summary$bic[6],col="red",cex=2,pch=20)
plot(regfit.full,scale="r2")
plot(regfit.full,scale="adjr2")
plot(regfit.full,scale="Cp")
plot(regfit.full,scale="bic")
coef(regfit.full,6)## (Intercept) AtBat Hits Walks CRBI
## 91.5117981 -1.8685892 7.6043976 3.6976468 0.6430169
## DivisionW PutOuts
## -122.9515338 0.2643076Subset Selection Methods: Forward and Backward Stepwise Selection
regfit.fwd=regsubsets(Salary~.,data=Hitters,nvmax=19,method="forward")
summary(regfit.fwd)## Subset selection object
## Call: regsubsets.formula(Salary ~ ., data = Hitters, nvmax = 19, method = "forward")
## 19 Variables (and intercept)
## Forced in Forced out
## AtBat FALSE FALSE
## Hits FALSE FALSE
## HmRun FALSE FALSE
## Runs FALSE FALSE
## RBI FALSE FALSE
## Walks FALSE FALSE
## Years FALSE FALSE
## CAtBat FALSE FALSE
## CHits FALSE FALSE
## CHmRun FALSE FALSE
## CRuns FALSE FALSE
## CRBI FALSE FALSE
## CWalks FALSE FALSE
## LeagueN FALSE FALSE
## DivisionW FALSE FALSE
## PutOuts FALSE FALSE
## Assists FALSE FALSE
## Errors FALSE FALSE
## NewLeagueN FALSE FALSE
## 1 subsets of each size up to 19
## Selection Algorithm: forward
## AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits CHmRun CRuns
## 1 ( 1 ) " " " " " " " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " "*" " " " " " " " " " " " " " " " " " "
## 5 ( 1 ) "*" "*" " " " " " " " " " " " " " " " " " "
## 6 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " " "
## 7 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " " "
## 8 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " "*"
## 9 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 10 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 11 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 12 ( 1 ) "*" "*" " " "*" " " "*" " " "*" " " " " "*"
## 13 ( 1 ) "*" "*" " " "*" " " "*" " " "*" " " " " "*"
## 14 ( 1 ) "*" "*" "*" "*" " " "*" " " "*" " " " " "*"
## 15 ( 1 ) "*" "*" "*" "*" " " "*" " " "*" "*" " " "*"
## 16 ( 1 ) "*" "*" "*" "*" "*" "*" " " "*" "*" " " "*"
## 17 ( 1 ) "*" "*" "*" "*" "*" "*" " " "*" "*" " " "*"
## 18 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" " " "*"
## 19 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"
## CRBI CWalks LeagueN DivisionW PutOuts Assists Errors NewLeagueN
## 1 ( 1 ) "*" " " " " " " " " " " " " " "
## 2 ( 1 ) "*" " " " " " " " " " " " " " "
## 3 ( 1 ) "*" " " " " " " "*" " " " " " "
## 4 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 5 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 6 ( 1 ) "*" " " " " "*" "*" " " " " " "
## 7 ( 1 ) "*" "*" " " "*" "*" " " " " " "
## 8 ( 1 ) "*" "*" " " "*" "*" " " " " " "
## 9 ( 1 ) "*" "*" " " "*" "*" " " " " " "
## 10 ( 1 ) "*" "*" " " "*" "*" "*" " " " "
## 11 ( 1 ) "*" "*" "*" "*" "*" "*" " " " "
## 12 ( 1 ) "*" "*" "*" "*" "*" "*" " " " "
## 13 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 14 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 15 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 16 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 17 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"
## 18 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"
## 19 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"regfit.bwd=regsubsets(Salary~.,data=Hitters,nvmax=19,method="backward")
summary(regfit.bwd)## Subset selection object
## Call: regsubsets.formula(Salary ~ ., data = Hitters, nvmax = 19, method = "backward")
## 19 Variables (and intercept)
## Forced in Forced out
## AtBat FALSE FALSE
## Hits FALSE FALSE
## HmRun FALSE FALSE
## Runs FALSE FALSE
## RBI FALSE FALSE
## Walks FALSE FALSE
## Years FALSE FALSE
## CAtBat FALSE FALSE
## CHits FALSE FALSE
## CHmRun FALSE FALSE
## CRuns FALSE FALSE
## CRBI FALSE FALSE
## CWalks FALSE FALSE
## LeagueN FALSE FALSE
## DivisionW FALSE FALSE
## PutOuts FALSE FALSE
## Assists FALSE FALSE
## Errors FALSE FALSE
## NewLeagueN FALSE FALSE
## 1 subsets of each size up to 19
## Selection Algorithm: backward
## AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits CHmRun CRuns
## 1 ( 1 ) " " " " " " " " " " " " " " " " " " " " "*"
## 2 ( 1 ) " " "*" " " " " " " " " " " " " " " " " "*"
## 3 ( 1 ) " " "*" " " " " " " " " " " " " " " " " "*"
## 4 ( 1 ) "*" "*" " " " " " " " " " " " " " " " " "*"
## 5 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " "*"
## 6 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " "*"
## 7 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " "*"
## 8 ( 1 ) "*" "*" " " " " " " "*" " " " " " " " " "*"
## 9 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 10 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 11 ( 1 ) "*" "*" " " " " " " "*" " " "*" " " " " "*"
## 12 ( 1 ) "*" "*" " " "*" " " "*" " " "*" " " " " "*"
## 13 ( 1 ) "*" "*" " " "*" " " "*" " " "*" " " " " "*"
## 14 ( 1 ) "*" "*" "*" "*" " " "*" " " "*" " " " " "*"
## 15 ( 1 ) "*" "*" "*" "*" " " "*" " " "*" "*" " " "*"
## 16 ( 1 ) "*" "*" "*" "*" "*" "*" " " "*" "*" " " "*"
## 17 ( 1 ) "*" "*" "*" "*" "*" "*" " " "*" "*" " " "*"
## 18 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" " " "*"
## 19 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"
## CRBI CWalks LeagueN DivisionW PutOuts Assists Errors NewLeagueN
## 1 ( 1 ) " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " "*" " " " " " "
## 4 ( 1 ) " " " " " " " " "*" " " " " " "
## 5 ( 1 ) " " " " " " " " "*" " " " " " "
## 6 ( 1 ) " " " " " " "*" "*" " " " " " "
## 7 ( 1 ) " " "*" " " "*" "*" " " " " " "
## 8 ( 1 ) "*" "*" " " "*" "*" " " " " " "
## 9 ( 1 ) "*" "*" " " "*" "*" " " " " " "
## 10 ( 1 ) "*" "*" " " "*" "*" "*" " " " "
## 11 ( 1 ) "*" "*" "*" "*" "*" "*" " " " "
## 12 ( 1 ) "*" "*" "*" "*" "*" "*" " " " "
## 13 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 14 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 15 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 16 ( 1 ) "*" "*" "*" "*" "*" "*" "*" " "
## 17 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"
## 18 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"
## 19 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*"coef(regfit.full,7)## (Intercept) Hits Walks CAtBat CHits
## 79.4509472 1.2833513 3.2274264 -0.3752350 1.4957073
## CHmRun DivisionW PutOuts
## 1.4420538 -129.9866432 0.2366813coef(regfit.fwd,7)## (Intercept) AtBat Hits Walks CRBI
## 109.7873062 -1.9588851 7.4498772 4.9131401 0.8537622
## CWalks DivisionW PutOuts
## -0.3053070 -127.1223928 0.2533404coef(regfit.bwd,7)## (Intercept) AtBat Hits Walks CRuns
## 105.6487488 -1.9762838 6.7574914 6.0558691 1.1293095
## CWalks DivisionW PutOuts
## -0.7163346 -116.1692169 0.3028847Subset Selection Methods: Choosing Among Models
set.seed(1)
train=sample(c(TRUE,FALSE), nrow(Hitters),rep=TRUE)
test=(!train)
regfit.best=regsubsets(Salary~.,data=Hitters[train,],nvmax=19)
test.mat=model.matrix(Salary~.,data=Hitters[test,])
val.errors=rep(NA,19)
for(i in 1:19){
coefi=coef(regfit.best,id=i)
pred=test.mat[,names(coefi)]%*%coefi
val.errors[i]=mean((Hitters$Salary[test]-pred)^2)
}
val.errors## [1] 220968.0 169157.1 178518.2 163426.1 168418.1 171270.6 162377.1
## [8] 157909.3 154055.7 148162.1 151156.4 151742.5 152214.5 157358.7
## [15] 158541.4 158743.3 159972.7 159859.8 160105.6which.min(val.errors)## [1] 10coef(regfit.best,10)## (Intercept) AtBat Hits Walks CAtBat CHits
## -80.2751499 -1.4683816 7.1625314 3.6430345 -0.1855698 1.1053238
## CHmRun CWalks LeagueN DivisionW PutOuts
## 1.3844863 -0.7483170 84.5576103 -53.0289658 0.2381662predict.regsubsets=function(object,newdata,id,...){
form=as.formula(object$call[[2]])
mat=model.matrix(form,newdata)
coefi=coef(object,id=id)
xvars=names(coefi)
mat[,xvars]%*%coefi
}
regfit.best=regsubsets(Salary~.,data=Hitters,nvmax=19)
coef(regfit.best,10)## (Intercept) AtBat Hits Walks CAtBat
## 162.5354420 -2.1686501 6.9180175 5.7732246 -0.1300798
## CRuns CRBI CWalks DivisionW PutOuts
## 1.4082490 0.7743122 -0.8308264 -112.3800575 0.2973726
## Assists
## 0.2831680k=10
set.seed(1)
folds=sample(1:k,nrow(Hitters),replace=TRUE)
cv.errors=matrix(NA,k,19, dimnames=list(NULL, paste(1:19)))
for(j in 1:k){
best.fit=regsubsets(Salary~.,data=Hitters[folds!=j,],nvmax=19)
for(i in 1:19){
pred=predict(best.fit,Hitters[folds==j,],id=i)
cv.errors[j,i]=mean( (Hitters$Salary[folds==j]-pred)^2)
}
}
mean.cv.errors=apply(cv.errors,2,mean)
mean.cv.errors## 1 2 3 4 5 6 7 8
## 160093.5 140196.8 153117.0 151159.3 146841.3 138302.6 144346.2 130207.7
## 9 10 11 12 13 14 15 16
## 129459.6 125334.7 125153.8 128273.5 133461.0 133974.6 131825.7 131882.8
## 17 18 19
## 132750.9 133096.2 132804.7par(mfrow=c(1,1))
plot(mean.cv.errors,type='b')
reg.best=regsubsets(Salary~.,data=Hitters, nvmax=19)
coef(reg.best,11)## (Intercept) AtBat Hits Walks CAtBat
## 135.7512195 -2.1277482 6.9236994 5.6202755 -0.1389914
## CRuns CRBI CWalks LeagueN DivisionW
## 1.4553310 0.7852528 -0.8228559 43.1116152 -111.1460252
## PutOuts Assists
## 0.2894087 0.2688277Ridge Regression and the Lasso
x=model.matrix(Salary~.,Hitters)[,-1]
y=Hitters$SalaryRidge Regression and the Lasso: Ridge Regression
library(glmnet)## Loading required package: Matrix
## Loading required package: foreach
## Loaded glmnet 2.0-2grid=10^seq(10,-2,length=100)
ridge.mod=glmnet(x,y,alpha=0,lambda=grid)
dim(coef(ridge.mod))## [1] 20 100ridge.mod$lambda[50]## [1] 11497.57coef(ridge.mod)[,50]## (Intercept) AtBat Hits HmRun Runs
## 407.356050200 0.036957182 0.138180344 0.524629976 0.230701523
## RBI Walks Years CAtBat CHits
## 0.239841459 0.289618741 1.107702929 0.003131815 0.011653637
## CHmRun CRuns CRBI CWalks LeagueN
## 0.087545670 0.023379882 0.024138320 0.025015421 0.085028114
## DivisionW PutOuts Assists Errors NewLeagueN
## -6.215440973 0.016482577 0.002612988 -0.020502690 0.301433531sqrt(sum(coef(ridge.mod)[-1,50]^2))## [1] 6.360612ridge.mod$lambda[60]## [1] 705.4802coef(ridge.mod)[,60]## (Intercept) AtBat Hits HmRun Runs
## 54.32519950 0.11211115 0.65622409 1.17980910 0.93769713
## RBI Walks Years CAtBat CHits
## 0.84718546 1.31987948 2.59640425 0.01083413 0.04674557
## CHmRun CRuns CRBI CWalks LeagueN
## 0.33777318 0.09355528 0.09780402 0.07189612 13.68370191
## DivisionW PutOuts Assists Errors NewLeagueN
## -54.65877750 0.11852289 0.01606037 -0.70358655 8.61181213sqrt(sum(coef(ridge.mod)[-1,60]^2))## [1] 57.11001predict(ridge.mod,s=50,type="coefficients")[1:20,]## (Intercept) AtBat Hits HmRun Runs
## 4.876610e+01 -3.580999e-01 1.969359e+00 -1.278248e+00 1.145892e+00
## RBI Walks Years CAtBat CHits
## 8.038292e-01 2.716186e+00 -6.218319e+00 5.447837e-03 1.064895e-01
## CHmRun CRuns CRBI CWalks LeagueN
## 6.244860e-01 2.214985e-01 2.186914e-01 -1.500245e-01 4.592589e+01
## DivisionW PutOuts Assists Errors NewLeagueN
## -1.182011e+02 2.502322e-01 1.215665e-01 -3.278600e+00 -9.496680e+00set.seed(1)
train=sample(1:nrow(x), nrow(x)/2)
test=(-train)
y.test=y[test]
ridge.mod=glmnet(x[train,],y[train],alpha=0,lambda=grid, thresh=1e-12)
ridge.pred=predict(ridge.mod,s=4,newx=x[test,])
mean((ridge.pred-y.test)^2)## [1] 101036.8mean((mean(y[train])-y.test)^2)## [1] 193253.1ridge.pred=predict(ridge.mod,s=1e10,newx=x[test,])
mean((ridge.pred-y.test)^2)## [1] 193253.1ridge.pred=predict(ridge.mod,s=0,newx=x[test,],exact=T)
mean((ridge.pred-y.test)^2)## [1] 114783.1lm(y~x, subset=train)##
## Call:
## lm(formula = y ~ x, subset = train)
##
## Coefficients:
## (Intercept) xAtBat xHits xHmRun xRuns
## 299.42849 -2.54027 8.36682 11.64512 -9.09923
## xRBI xWalks xYears xCAtBat xCHits
## 2.44105 9.23440 -22.93673 -0.18154 -0.11598
## xCHmRun xCRuns xCRBI xCWalks xLeagueN
## -1.33888 3.32838 0.07536 -1.07841 59.76065
## xDivisionW xPutOuts xAssists xErrors xNewLeagueN
## -98.86233 0.34087 0.34165 -0.64207 -0.67442predict(ridge.mod,s=0,exact=T,type="coefficients")[1:20,]## (Intercept) AtBat Hits HmRun Runs
## 299.42883596 -2.54014665 8.36611719 11.64400720 -9.09877719
## RBI Walks Years CAtBat CHits
## 2.44152119 9.23403909 -22.93584442 -0.18160843 -0.11561496
## CHmRun CRuns CRBI CWalks LeagueN
## -1.33836534 3.32817777 0.07511771 -1.07828647 59.76529059
## DivisionW PutOuts Assists Errors NewLeagueN
## -98.85996590 0.34086400 0.34165605 -0.64205839 -0.67606314set.seed(1)
cv.out=cv.glmnet(x[train,],y[train],alpha=0)
plot(cv.out)
bestlam=cv.out$lambda.min
bestlam## [1] 211.7416ridge.pred=predict(ridge.mod,s=bestlam,newx=x[test,])
mean((ridge.pred-y.test)^2)## [1] 96015.51out=glmnet(x,y,alpha=0)
predict(out,type="coefficients",s=bestlam)[1:20,]## (Intercept) AtBat Hits HmRun Runs
## 9.88487157 0.03143991 1.00882875 0.13927624 1.11320781
## RBI Walks Years CAtBat CHits
## 0.87318990 1.80410229 0.13074381 0.01113978 0.06489843
## CHmRun CRuns CRBI CWalks LeagueN
## 0.45158546 0.12900049 0.13737712 0.02908572 27.18227535
## DivisionW PutOuts Assists Errors NewLeagueN
## -91.63411299 0.19149252 0.04254536 -1.81244470 7.21208390Ridge Regression and the Lasso: The Lasso
lasso.mod=glmnet(x[train,],y[train],alpha=1,lambda=grid)
plot(lasso.mod)
set.seed(1)
cv.out=cv.glmnet(x[train,],y[train],alpha=1)
plot(cv.out)
bestlam=cv.out$lambda.min
lasso.pred=predict(lasso.mod,s=bestlam,newx=x[test,])
mean((lasso.pred-y.test)^2)## [1] 100743.4out=glmnet(x,y,alpha=1,lambda=grid)
lasso.coef=predict(out,type="coefficients",s=bestlam)[1:20,]
lasso.coef## (Intercept) AtBat Hits HmRun Runs
## 18.5394844 0.0000000 1.8735390 0.0000000 0.0000000
## RBI Walks Years CAtBat CHits
## 0.0000000 2.2178444 0.0000000 0.0000000 0.0000000
## CHmRun CRuns CRBI CWalks LeagueN
## 0.0000000 0.2071252 0.4130132 0.0000000 3.2666677
## DivisionW PutOuts Assists Errors NewLeagueN
## -103.4845458 0.2204284 0.0000000 0.0000000 0.0000000lasso.coef[lasso.coef!=0]## (Intercept) Hits Walks CRuns CRBI
## 18.5394844 1.8735390 2.2178444 0.2071252 0.4130132
## LeagueN DivisionW PutOuts
## 3.2666677 -103.4845458 0.2204284PCR and PLS Regression : Principal Components Regression
library(pls)##
## Attaching package: 'pls'
##
## The following object is masked from 'package:stats':
##
## loadingsset.seed(2)
pcr.fit=pcr(Salary~., data=Hitters,scale=TRUE,validation="CV")
summary(pcr.fit)## Data: X dimension: 263 19
## Y dimension: 263 1
## Fit method: svdpc
## Number of components considered: 19
##
## VALIDATION: RMSEP
## Cross-validated using 10 random segments.
## (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## CV 452 348.9 352.2 353.5 352.8 350.1 349.1
## adjCV 452 348.7 351.8 352.9 352.1 349.3 348.0
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps
## CV 349.6 350.9 352.9 353.8 355.0 356.2 363.5
## adjCV 348.5 349.8 351.6 352.3 353.4 354.5 361.6
## 14 comps 15 comps 16 comps 17 comps 18 comps 19 comps
## CV 355.2 357.4 347.6 350.1 349.2 352.6
## adjCV 352.8 355.2 345.5 347.6 346.7 349.8
##
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 38.31 60.16 70.84 79.03 84.29 88.63 92.26
## Salary 40.63 41.58 42.17 43.22 44.90 46.48 46.69
## 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps 14 comps
## X 94.96 96.28 97.26 97.98 98.65 99.15 99.47
## Salary 46.75 46.86 47.76 47.82 47.85 48.10 50.40
## 15 comps 16 comps 17 comps 18 comps 19 comps
## X 99.75 99.89 99.97 99.99 100.00
## Salary 50.55 53.01 53.85 54.61 54.61validationplot(pcr.fit,val.type="MSEP")
set.seed(1)
pcr.fit=pcr(Salary~., data=Hitters,subset=train,scale=TRUE, validation="CV")
validationplot(pcr.fit,val.type="MSEP")
pcr.pred=predict(pcr.fit,x[test,],ncomp=7)
mean((pcr.pred-y.test)^2)## [1] 96556.22pcr.fit=pcr(y~x,scale=TRUE,ncomp=7)
summary(pcr.fit)## Data: X dimension: 263 19
## Y dimension: 263 1
## Fit method: svdpc
## Number of components considered: 7
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 38.31 60.16 70.84 79.03 84.29 88.63 92.26
## y 40.63 41.58 42.17 43.22 44.90 46.48 46.69PCR and PLS Regression : Partial Least Squares
set.seed(1)
pls.fit=plsr(Salary~., data=Hitters,subset=train,scale=TRUE, validation="CV")
summary(pls.fit)## Data: X dimension: 131 19
## Y dimension: 131 1
## Fit method: kernelpls
## Number of components considered: 19
##
## VALIDATION: RMSEP
## Cross-validated using 10 random segments.
## (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## CV 464.6 394.2 391.5 393.1 395.0 415.0 424.0
## adjCV 464.6 393.4 390.2 391.1 392.9 411.5 418.8
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps
## CV 424.5 415.8 404.6 407.1 412.0 414.4 410.3
## adjCV 418.9 411.4 400.7 402.2 407.2 409.3 405.6
## 14 comps 15 comps 16 comps 17 comps 18 comps 19 comps
## CV 406.2 408.6 410.5 408.8 407.8 410.2
## adjCV 401.8 403.9 405.6 404.1 403.2 405.5
##
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 38.12 53.46 66.05 74.49 79.33 84.56 87.09
## Salary 33.58 38.96 41.57 42.43 44.04 45.59 47.05
## 8 comps 9 comps 10 comps 11 comps 12 comps 13 comps 14 comps
## X 90.74 92.55 93.94 97.23 97.88 98.35 98.85
## Salary 47.53 48.42 49.68 50.04 50.54 50.78 50.92
## 15 comps 16 comps 17 comps 18 comps 19 comps
## X 99.11 99.43 99.78 99.99 100.00
## Salary 51.04 51.11 51.15 51.16 51.18validationplot(pls.fit,val.type="MSEP")
pls.pred=predict(pls.fit,x[test,],ncomp=2)
mean((pls.pred-y.test)^2)## [1] 101417.5pls.fit=plsr(Salary~., data=Hitters,scale=TRUE,ncomp=2)
summary(pls.fit)## Data: X dimension: 263 19
## Y dimension: 263 1
## Fit method: kernelpls
## Number of components considered: 2
## TRAINING: % variance explained
## 1 comps 2 comps
## X 38.08 51.03
## Salary 43.05 46.40Chapter 7: Non-linear Modeling
attach(Wage)## The following object is masked from Auto:
##
## year
##
## The following object is masked from Boston:
##
## agePolynomial Regression and Step Functions
fit=lm(wage~poly(age,4),data=Wage)
coef(summary(fit))## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 111.70361 0.7287409 153.283015 0.000000e+00
## poly(age, 4)1 447.06785 39.9147851 11.200558 1.484604e-28
## poly(age, 4)2 -478.31581 39.9147851 -11.983424 2.355831e-32
## poly(age, 4)3 125.52169 39.9147851 3.144742 1.678622e-03
## poly(age, 4)4 -77.91118 39.9147851 -1.951938 5.103865e-02fit2=lm(wage~poly(age,4,raw=T),data=Wage)
coef(summary(fit2))## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.841542e+02 6.004038e+01 -3.067172 0.0021802539
## poly(age, 4, raw = T)1 2.124552e+01 5.886748e+00 3.609042 0.0003123618
## poly(age, 4, raw = T)2 -5.638593e-01 2.061083e-01 -2.735743 0.0062606446
## poly(age, 4, raw = T)3 6.810688e-03 3.065931e-03 2.221409 0.0263977518
## poly(age, 4, raw = T)4 -3.203830e-05 1.641359e-05 -1.951938 0.0510386498fit2a=lm(wage~age+I(age^2)+I(age^3)+I(age^4),data=Wage)
coef(fit2a)## (Intercept) age I(age^2) I(age^3) I(age^4)
## -1.841542e+02 2.124552e+01 -5.638593e-01 6.810688e-03 -3.203830e-05fit2b=lm(wage~cbind(age,age^2,age^3,age^4),data=Wage)
agelims=range(age)
age.grid=seq(from=agelims[1],to=agelims[2])
preds=predict(fit,newdata=list(age=age.grid),se=TRUE)
se.bands=cbind(preds$fit+2*preds$se.fit,preds$fit-2*preds$se.fit)
par(mfrow=c(1,2),mar=c(4.5,4.5,1,1),oma=c(0,0,4,0))
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Degree-4 Polynomial",outer=T)
lines(age.grid,preds$fit,lwd=2,col="blue")
matlines(age.grid,se.bands,lwd=1,col="blue",lty=3)
preds2=predict(fit2,newdata=list(age=age.grid),se=TRUE)
max(abs(preds$fit-preds2$fit))## [1] 7.81597e-11fit.1=lm(wage~age,data=Wage)
fit.2=lm(wage~poly(age,2),data=Wage)
fit.3=lm(wage~poly(age,3),data=Wage)
fit.4=lm(wage~poly(age,4),data=Wage)
fit.5=lm(wage~poly(age,5),data=Wage)
anova(fit.1,fit.2,fit.3,fit.4,fit.5)## Analysis of Variance Table
##
## Model 1: wage ~ age
## Model 2: wage ~ poly(age, 2)
## Model 3: wage ~ poly(age, 3)
## Model 4: wage ~ poly(age, 4)
## Model 5: wage ~ poly(age, 5)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2998 5022216
## 2 2997 4793430 1 228786 143.5931 < 2.2e-16 ***
## 3 2996 4777674 1 15756 9.8888 0.001679 **
## 4 2995 4771604 1 6070 3.8098 0.051046 .
## 5 2994 4770322 1 1283 0.8050 0.369682
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coef(summary(fit.5))## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 111.70361 0.7287647 153.2780243 0.000000e+00
## poly(age, 5)1 447.06785 39.9160847 11.2001930 1.491111e-28
## poly(age, 5)2 -478.31581 39.9160847 -11.9830341 2.367734e-32
## poly(age, 5)3 125.52169 39.9160847 3.1446392 1.679213e-03
## poly(age, 5)4 -77.91118 39.9160847 -1.9518743 5.104623e-02
## poly(age, 5)5 -35.81289 39.9160847 -0.8972045 3.696820e-01(-11.983)^2## [1] 143.5923fit.1=lm(wage~education+age,data=Wage)
fit.2=lm(wage~education+poly(age,2),data=Wage)
fit.3=lm(wage~education+poly(age,3),data=Wage)
anova(fit.1,fit.2,fit.3)## Analysis of Variance Table
##
## Model 1: wage ~ education + age
## Model 2: wage ~ education + poly(age, 2)
## Model 3: wage ~ education + poly(age, 3)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 2994 3867992
## 2 2993 3725395 1 142597 114.6969 <2e-16 ***
## 3 2992 3719809 1 5587 4.4936 0.0341 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit=glm(I(wage>250)~poly(age,4),data=Wage,family=binomial)
preds=predict(fit,newdata=list(age=age.grid),se=T)
pfit=exp(preds$fit)/(1+exp(preds$fit))
se.bands.logit = cbind(preds$fit+2*preds$se.fit, preds$fit-2*preds$se.fit)
se.bands = exp(se.bands.logit)/(1+exp(se.bands.logit))
preds=predict(fit,newdata=list(age=age.grid),type="response",se=T)
plot(age,I(wage>250),xlim=agelims,type="n",ylim=c(0,.2))
points(jitter(age), I((wage>250)/5),cex=.5,pch="|",col="darkgrey")
lines(age.grid,pfit,lwd=2, col="blue")
matlines(age.grid,se.bands,lwd=1,col="blue",lty=3)
table(cut(age,4))##
## (17.9,33.5] (33.5,49] (49,64.5] (64.5,80.1]
## 750 1399 779 72fit=lm(wage~cut(age,4),data=Wage)
coef(summary(fit))## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 94.158392 1.476069 63.789970 0.000000e+00
## cut(age, 4)(33.5,49] 24.053491 1.829431 13.148074 1.982315e-38
## cut(age, 4)(49,64.5] 23.664559 2.067958 11.443444 1.040750e-29
## cut(age, 4)(64.5,80.1] 7.640592 4.987424 1.531972 1.256350e-01Splines
library(splines)
fit=lm(wage~bs(age,knots=c(25,40,60)),data=Wage)
pred=predict(fit,newdata=list(age=age.grid),se=T)
plot(age,wage,col="gray")
lines(age.grid,pred$fit,lwd=2)
lines(age.grid,pred$fit+2*pred$se,lty="dashed")
lines(age.grid,pred$fit-2*pred$se,lty="dashed")
dim(bs(age,knots=c(25,40,60)))## [1] 3000 6dim(bs(age,df=6))## [1] 3000 6attr(bs(age,df=6),"knots")## 25% 50% 75%
## 33.75 42.00 51.00fit2=lm(wage~ns(age,df=4),data=Wage)
pred2=predict(fit2,newdata=list(age=age.grid),se=T)
lines(age.grid, pred2$fit,col="red",lwd=2)
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Smoothing Spline")
fit=smooth.spline(age,wage,df=16)
fit2=smooth.spline(age,wage,cv=TRUE)## Warning in smooth.spline(age, wage, cv = TRUE): cross-validation with non-
## unique 'x' values seems doubtfulfit2$df## [1] 6.794596lines(fit,col="red",lwd=2)
lines(fit2,col="blue",lwd=2)
legend("topright",legend=c("16 DF","6.8 DF"),col=c("red","blue"),lty=1,lwd=2,cex=.8)
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Local Regression")
fit=loess(wage~age,span=.2,data=Wage)
fit2=loess(wage~age,span=.5,data=Wage)
lines(age.grid,predict(fit,data.frame(age=age.grid)),col="red",lwd=2)
lines(age.grid,predict(fit2,data.frame(age=age.grid)),col="blue",lwd=2)
legend("topright",legend=c("Span=0.2","Span=0.5"),col=c("red","blue"),lty=1,lwd=2,cex=.8)
GAMs
Hmm, have to comment out some plots here as they are not working!
names(Wage)## [1] "year" "age" "sex" "maritl" "race"
## [6] "education" "region" "jobclass" "health" "health_ins"
## [11] "logwage" "wage"gam1=lm(wage~ns(year,4)+ns(age,5)+education,data=Wage)
#library(mgcv)
library(gam)## Loaded gam 1.12gam.m3=gam(wage~s(year,4)+s(age,5)+education,data=Wage)
summary(gam.m3)##
## Call: gam(formula = wage ~ s(year, 4) + s(age, 5) + education, data = Wage)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -119.43 -19.70 -3.33 14.17 213.48
##
## (Dispersion Parameter for gaussian family taken to be 1235.69)
##
## Null Deviance: 5222086 on 2999 degrees of freedom
## Residual Deviance: 3689770 on 2986 degrees of freedom
## AIC: 29887.75
##
## Number of Local Scoring Iterations: 2
##
## Anova for Parametric Effects
## Df Sum Sq Mean Sq F value Pr(>F)
## s(year, 4) 1 27162 27162 21.981 2.877e-06 ***
## s(age, 5) 1 195338 195338 158.081 < 2.2e-16 ***
## education 4 1069726 267432 216.423 < 2.2e-16 ***
## Residuals 2986 3689770 1236
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Anova for Nonparametric Effects
## Npar Df Npar F Pr(F)
## (Intercept)
## s(year, 4) 3 1.086 0.3537
## s(age, 5) 4 32.380 <2e-16 ***
## education
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1par(mfrow=c(1,3))
#plot.gam(gam.m3, se=TRUE,col="blue")
plot.gam(gam1, se=TRUE, col="red")
gam.m1=gam(wage~s(age,5)+education,data=Wage)
gam.m2=gam(wage~year+s(age,5)+education,data=Wage)
anova(gam.m1,gam.m2,gam.m3,test="F")## Analysis of Deviance Table
##
## Model 1: wage ~ s(age, 5) + education
## Model 2: wage ~ year + s(age, 5) + education
## Model 3: wage ~ s(year, 4) + s(age, 5) + education
## Resid. Df Resid. Dev Df Deviance F Pr(>F)
## 1 2990 3711731
## 2 2989 3693842 1 17889.2 14.4771 0.0001447 ***
## 3 2986 3689770 3 4071.1 1.0982 0.3485661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(gam.m3)##
## Call: gam(formula = wage ~ s(year, 4) + s(age, 5) + education, data = Wage)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -119.43 -19.70 -3.33 14.17 213.48
##
## (Dispersion Parameter for gaussian family taken to be 1235.69)
##
## Null Deviance: 5222086 on 2999 degrees of freedom
## Residual Deviance: 3689770 on 2986 degrees of freedom
## AIC: 29887.75
##
## Number of Local Scoring Iterations: 2
##
## Anova for Parametric Effects
## Df Sum Sq Mean Sq F value Pr(>F)
## s(year, 4) 1 27162 27162 21.981 2.877e-06 ***
## s(age, 5) 1 195338 195338 158.081 < 2.2e-16 ***
## education 4 1069726 267432 216.423 < 2.2e-16 ***
## Residuals 2986 3689770 1236
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Anova for Nonparametric Effects
## Npar Df Npar F Pr(F)
## (Intercept)
## s(year, 4) 3 1.086 0.3537
## s(age, 5) 4 32.380 <2e-16 ***
## education
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1preds=predict(gam.m2,newdata=Wage)
gam.lo=gam(wage~s(year,df=4)+lo(age,span=0.7)+education,data=Wage)
#plot.gam(gam.lo, se=TRUE, col="green")
gam.lo.i=gam(wage~lo(year,age,span=0.5)+education,data=Wage)
library(akima)
#plot(gam.lo.i)
gam.lr=gam(I(wage>250)~year+s(age,df=5)+education,family=binomial,data=Wage)
par(mfrow=c(1,3))
#plot(gam.lr,se=T,col="green")
table(education,I(wage>250))##
## education FALSE TRUE
## 1. < HS Grad 268 0
## 2. HS Grad 966 5
## 3. Some College 643 7
## 4. College Grad 663 22
## 5. Advanced Degree 381 45gam.lr.s=gam(I(wage>250)~year+s(age,df=5)+education,family=binomial,data=Wage,subset=(education!="1. < HS Grad"))
#plot(gam.lr.s,se=T,col="green")Chapter 8: Decision Trees
Fitting Classification Trees
library(tree)
attach(Carseats)## The following objects are masked from Carseats (pos = 17):
##
## Advertising, Age, CompPrice, Education, Income, Population,
## Price, Sales, ShelveLoc, Urban, USHigh=ifelse(Sales<=8,"No","Yes")
Carseats=data.frame(Carseats,High)
tree.carseats=tree(High~.-Sales,Carseats)
summary(tree.carseats)##
## Classification tree:
## tree(formula = High ~ . - Sales, data = Carseats)
## Variables actually used in tree construction:
## [1] "ShelveLoc" "Price" "Income" "CompPrice" "Population"
## [6] "Advertising" "Age" "US"
## Number of terminal nodes: 27
## Residual mean deviance: 0.4575 = 170.7 / 373
## Misclassification error rate: 0.09 = 36 / 400plot(tree.carseats)
text(tree.carseats,pretty=0)
tree.carseats## node), split, n, deviance, yval, (yprob)
## * denotes terminal node
##
## 1) root 400 541.500 No ( 0.59000 0.41000 )
## 2) ShelveLoc: Bad,Medium 315 390.600 No ( 0.68889 0.31111 )
## 4) Price < 92.5 46 56.530 Yes ( 0.30435 0.69565 )
## 8) Income < 57 10 12.220 No ( 0.70000 0.30000 )
## 16) CompPrice < 110.5 5 0.000 No ( 1.00000 0.00000 ) *
## 17) CompPrice > 110.5 5 6.730 Yes ( 0.40000 0.60000 ) *
## 9) Income > 57 36 35.470 Yes ( 0.19444 0.80556 )
## 18) Population < 207.5 16 21.170 Yes ( 0.37500 0.62500 ) *
## 19) Population > 207.5 20 7.941 Yes ( 0.05000 0.95000 ) *
## 5) Price > 92.5 269 299.800 No ( 0.75465 0.24535 )
## 10) Advertising < 13.5 224 213.200 No ( 0.81696 0.18304 )
## 20) CompPrice < 124.5 96 44.890 No ( 0.93750 0.06250 )
## 40) Price < 106.5 38 33.150 No ( 0.84211 0.15789 )
## 80) Population < 177 12 16.300 No ( 0.58333 0.41667 )
## 160) Income < 60.5 6 0.000 No ( 1.00000 0.00000 ) *
## 161) Income > 60.5 6 5.407 Yes ( 0.16667 0.83333 ) *
## 81) Population > 177 26 8.477 No ( 0.96154 0.03846 ) *
## 41) Price > 106.5 58 0.000 No ( 1.00000 0.00000 ) *
## 21) CompPrice > 124.5 128 150.200 No ( 0.72656 0.27344 )
## 42) Price < 122.5 51 70.680 Yes ( 0.49020 0.50980 )
## 84) ShelveLoc: Bad 11 6.702 No ( 0.90909 0.09091 ) *
## 85) ShelveLoc: Medium 40 52.930 Yes ( 0.37500 0.62500 )
## 170) Price < 109.5 16 7.481 Yes ( 0.06250 0.93750 ) *
## 171) Price > 109.5 24 32.600 No ( 0.58333 0.41667 )
## 342) Age < 49.5 13 16.050 Yes ( 0.30769 0.69231 ) *
## 343) Age > 49.5 11 6.702 No ( 0.90909 0.09091 ) *
## 43) Price > 122.5 77 55.540 No ( 0.88312 0.11688 )
## 86) CompPrice < 147.5 58 17.400 No ( 0.96552 0.03448 ) *
## 87) CompPrice > 147.5 19 25.010 No ( 0.63158 0.36842 )
## 174) Price < 147 12 16.300 Yes ( 0.41667 0.58333 )
## 348) CompPrice < 152.5 7 5.742 Yes ( 0.14286 0.85714 ) *
## 349) CompPrice > 152.5 5 5.004 No ( 0.80000 0.20000 ) *
## 175) Price > 147 7 0.000 No ( 1.00000 0.00000 ) *
## 11) Advertising > 13.5 45 61.830 Yes ( 0.44444 0.55556 )
## 22) Age < 54.5 25 25.020 Yes ( 0.20000 0.80000 )
## 44) CompPrice < 130.5 14 18.250 Yes ( 0.35714 0.64286 )
## 88) Income < 100 9 12.370 No ( 0.55556 0.44444 ) *
## 89) Income > 100 5 0.000 Yes ( 0.00000 1.00000 ) *
## 45) CompPrice > 130.5 11 0.000 Yes ( 0.00000 1.00000 ) *
## 23) Age > 54.5 20 22.490 No ( 0.75000 0.25000 )
## 46) CompPrice < 122.5 10 0.000 No ( 1.00000 0.00000 ) *
## 47) CompPrice > 122.5 10 13.860 No ( 0.50000 0.50000 )
## 94) Price < 125 5 0.000 Yes ( 0.00000 1.00000 ) *
## 95) Price > 125 5 0.000 No ( 1.00000 0.00000 ) *
## 3) ShelveLoc: Good 85 90.330 Yes ( 0.22353 0.77647 )
## 6) Price < 135 68 49.260 Yes ( 0.11765 0.88235 )
## 12) US: No 17 22.070 Yes ( 0.35294 0.64706 )
## 24) Price < 109 8 0.000 Yes ( 0.00000 1.00000 ) *
## 25) Price > 109 9 11.460 No ( 0.66667 0.33333 ) *
## 13) US: Yes 51 16.880 Yes ( 0.03922 0.96078 ) *
## 7) Price > 135 17 22.070 No ( 0.64706 0.35294 )
## 14) Income < 46 6 0.000 No ( 1.00000 0.00000 ) *
## 15) Income > 46 11 15.160 Yes ( 0.45455 0.54545 ) *set.seed(2)
train=sample(1:nrow(Carseats), 200)
Carseats.test=Carseats[-train,]
High.test=High[-train]
tree.carseats=tree(High~.-Sales,Carseats,subset=train)
tree.pred=predict(tree.carseats,Carseats.test,type="class")
table(tree.pred,High.test)## High.test
## tree.pred No Yes
## No 86 27
## Yes 30 57(86+57)/200## [1] 0.715set.seed(3)
cv.carseats=cv.tree(tree.carseats,FUN=prune.misclass)
names(cv.carseats)## [1] "size" "dev" "k" "method"cv.carseats## $size
## [1] 19 17 14 13 9 7 3 2 1
##
## $dev
## [1] 55 55 53 52 50 56 69 65 80
##
## $k
## [1] -Inf 0.0000000 0.6666667 1.0000000 1.7500000 2.0000000
## [7] 4.2500000 5.0000000 23.0000000
##
## $method
## [1] "misclass"
##
## attr(,"class")
## [1] "prune" "tree.sequence"par(mfrow=c(1,2))
plot(cv.carseats$size,cv.carseats$dev,type="b")
plot(cv.carseats$k,cv.carseats$dev,type="b")
prune.carseats=prune.misclass(tree.carseats,best=9)
plot(prune.carseats)
text(prune.carseats,pretty=0)
tree.pred=predict(prune.carseats,Carseats.test,type="class")
table(tree.pred,High.test)## High.test
## tree.pred No Yes
## No 94 24
## Yes 22 60(94+60)/200## [1] 0.77prune.carseats=prune.misclass(tree.carseats,best=15)
plot(prune.carseats)
text(prune.carseats,pretty=0)
tree.pred=predict(prune.carseats,Carseats.test,type="class")
table(tree.pred,High.test)## High.test
## tree.pred No Yes
## No 86 22
## Yes 30 62(86+62)/200## [1] 0.74Fitting Regression Trees
#MASS
set.seed(1)
train = sample(1:nrow(Boston), nrow(Boston)/2)
tree.boston=tree(medv~.,Boston,subset=train)
summary(tree.boston)##
## Regression tree:
## tree(formula = medv ~ ., data = Boston, subset = train)
## Variables actually used in tree construction:
## [1] "lstat" "rm" "dis"
## Number of terminal nodes: 8
## Residual mean deviance: 12.65 = 3099 / 245
## Distribution of residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -14.10000 -2.04200 -0.05357 0.00000 1.96000 12.60000plot(tree.boston)
text(tree.boston,pretty=0)
cv.boston=cv.tree(tree.boston)
plot(cv.boston$size,cv.boston$dev,type='b')
prune.boston=prune.tree(tree.boston,best=5)
plot(prune.boston)
text(prune.boston,pretty=0)
yhat=predict(tree.boston,newdata=Boston[-train,])
boston.test=Boston[-train,"medv"]
plot(yhat,boston.test)
abline(0,1)
mean((yhat-boston.test)^2)## [1] 25.04559Bagging and Random Forests
library(randomForest)## randomForest 4.6-12
## Type rfNews() to see new features/changes/bug fixes.set.seed(1)
bag.boston=randomForest(medv~.,data=Boston,subset=train,mtry=13,importance=TRUE)
bag.boston##
## Call:
## randomForest(formula = medv ~ ., data = Boston, mtry = 13, importance = TRUE, subset = train)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 13
##
## Mean of squared residuals: 11.02509
## % Var explained: 86.65yhat.bag = predict(bag.boston,newdata=Boston[-train,])
plot(yhat.bag, boston.test)
abline(0,1)
mean((yhat.bag-boston.test)^2)## [1] 13.47349bag.boston=randomForest(medv~.,data=Boston,subset=train,mtry=13,ntree=25)
yhat.bag = predict(bag.boston,newdata=Boston[-train,])
mean((yhat.bag-boston.test)^2)## [1] 13.43068set.seed(1)
rf.boston=randomForest(medv~.,data=Boston,subset=train,mtry=6,importance=TRUE)
yhat.rf = predict(rf.boston,newdata=Boston[-train,])
mean((yhat.rf-boston.test)^2)## [1] 11.48022importance(rf.boston)## %IncMSE IncNodePurity
## crim 12.547772 1094.65382
## zn 1.375489 64.40060
## indus 9.304258 1086.09103
## chas 2.518766 76.36804
## nox 12.835614 1008.73703
## rm 31.646147 6705.02638
## age 9.970243 575.13702
## dis 12.774430 1351.01978
## rad 3.911852 93.78200
## tax 7.624043 453.19472
## ptratio 12.008194 919.06760
## black 7.376024 358.96935
## lstat 27.666896 6927.98475varImpPlot(rf.boston)
Boosting
library(gbm)## Loading required package: survival
##
## Attaching package: 'survival'
##
## The following object is masked from 'package:boot':
##
## aml
##
## Loading required package: lattice
##
## Attaching package: 'lattice'
##
## The following object is masked from 'package:boot':
##
## melanoma
##
## Loading required package: parallel
## Loaded gbm 2.1.1set.seed(1)
boost.boston=gbm(medv~.,data=Boston[train,],distribution="gaussian",n.trees=5000,interaction.depth=4)
summary(boost.boston)
## var rel.inf
## lstat lstat 45.9627334
## rm rm 31.2238187
## dis dis 6.8087398
## crim crim 4.0743784
## nox nox 2.5605001
## ptratio ptratio 2.2748652
## black black 1.7971159
## age age 1.6488532
## tax tax 1.3595005
## indus indus 1.2705924
## chas chas 0.8014323
## rad rad 0.2026619
## zn zn 0.0148083par(mfrow=c(1,2))
plot(boost.boston,i="rm")
plot(boost.boston,i="lstat")
yhat.boost=predict(boost.boston,newdata=Boston[-train,],n.trees=5000)
mean((yhat.boost-boston.test)^2)## [1] 11.84434boost.boston=gbm(medv~.,data=Boston[train,],distribution="gaussian",n.trees=5000,interaction.depth=4,shrinkage=0.2,verbose=F)
yhat.boost=predict(boost.boston,newdata=Boston[-train,],n.trees=5000)
mean((yhat.boost-boston.test)^2)## [1] 11.51109Chapter 9: Support Vector Machines
Support Vector Classifier
set.seed(1)
x=matrix(rnorm(20*2), ncol=2)
y=c(rep(-1,10), rep(1,10))
x[y==1,]=x[y==1,] + 1
plot(x, col=(3-y))
dat=data.frame(x=x, y=as.factor(y))
library(e1071)
svmfit=svm(y~., data=dat, kernel="linear", cost=10,scale=FALSE)
plot(svmfit, dat)
svmfit$index## [1] 1 2 5 7 14 16 17summary(svmfit)##
## Call:
## svm(formula = y ~ ., data = dat, kernel = "linear", cost = 10,
## scale = FALSE)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 10
## gamma: 0.5
##
## Number of Support Vectors: 7
##
## ( 4 3 )
##
##
## Number of Classes: 2
##
## Levels:
## -1 1svmfit=svm(y~., data=dat, kernel="linear", cost=0.1,scale=FALSE)
plot(svmfit, dat)
svmfit$index## [1] 1 2 3 4 5 7 9 10 12 13 14 15 16 17 18 20set.seed(1)
tune.out=tune(svm,y~.,data=dat,kernel="linear",ranges=list(cost=c(0.001, 0.01, 0.1, 1,5,10,100)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 0.1
##
## - best performance: 0.1
##
## - Detailed performance results:
## cost error dispersion
## 1 1e-03 0.70 0.4216370
## 2 1e-02 0.70 0.4216370
## 3 1e-01 0.10 0.2108185
## 4 1e+00 0.15 0.2415229
## 5 5e+00 0.15 0.2415229
## 6 1e+01 0.15 0.2415229
## 7 1e+02 0.15 0.2415229bestmod=tune.out$best.model
summary(bestmod)##
## Call:
## best.tune(method = svm, train.x = y ~ ., data = dat, ranges = list(cost = c(0.001,
## 0.01, 0.1, 1, 5, 10, 100)), kernel = "linear")
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 0.1
## gamma: 0.5
##
## Number of Support Vectors: 16
##
## ( 8 8 )
##
##
## Number of Classes: 2
##
## Levels:
## -1 1xtest=matrix(rnorm(20*2), ncol=2)
ytest=sample(c(-1,1), 20, rep=TRUE)
xtest[ytest==1,]=xtest[ytest==1,] + 1
testdat=data.frame(x=xtest, y=as.factor(ytest))
ypred=predict(bestmod,testdat)
table(predict=ypred, truth=testdat$y)## truth
## predict -1 1
## -1 11 1
## 1 0 8svmfit=svm(y~., data=dat, kernel="linear", cost=.01,scale=FALSE)
ypred=predict(svmfit,testdat)
table(predict=ypred, truth=testdat$y)## truth
## predict -1 1
## -1 11 2
## 1 0 7x[y==1,]=x[y==1,]+0.5
plot(x, col=(y+5)/2, pch=19)
dat=data.frame(x=x,y=as.factor(y))
svmfit=svm(y~., data=dat, kernel="linear", cost=1e5)
summary(svmfit)##
## Call:
## svm(formula = y ~ ., data = dat, kernel = "linear", cost = 1e+05)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 1e+05
## gamma: 0.5
##
## Number of Support Vectors: 3
##
## ( 1 2 )
##
##
## Number of Classes: 2
##
## Levels:
## -1 1plot(svmfit, dat)
svmfit=svm(y~., data=dat, kernel="linear", cost=1)
summary(svmfit)##
## Call:
## svm(formula = y ~ ., data = dat, kernel = "linear", cost = 1)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 1
## gamma: 0.5
##
## Number of Support Vectors: 7
##
## ( 4 3 )
##
##
## Number of Classes: 2
##
## Levels:
## -1 1plot(svmfit,dat)
Support Vector Machine
set.seed(1)
x=matrix(rnorm(200*2), ncol=2)
x[1:100,]=x[1:100,]+2
x[101:150,]=x[101:150,]-2
y=c(rep(1,150),rep(2,50))
dat=data.frame(x=x,y=as.factor(y))
plot(x, col=y)
train=sample(200,100)
svmfit=svm(y~., data=dat[train,], kernel="radial", gamma=1, cost=1)
plot(svmfit, dat[train,])
summary(svmfit)##
## Call:
## svm(formula = y ~ ., data = dat[train, ], kernel = "radial",
## gamma = 1, cost = 1)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: radial
## cost: 1
## gamma: 1
##
## Number of Support Vectors: 37
##
## ( 17 20 )
##
##
## Number of Classes: 2
##
## Levels:
## 1 2svmfit=svm(y~., data=dat[train,], kernel="radial",gamma=1,cost=1e5)
plot(svmfit,dat[train,])
set.seed(1)
tune.out=tune(svm, y~., data=dat[train,], kernel="radial", ranges=list(cost=c(0.1,1,10,100,1000),gamma=c(0.5,1,2,3,4)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost gamma
## 1 2
##
## - best performance: 0.12
##
## - Detailed performance results:
## cost gamma error dispersion
## 1 1e-01 0.5 0.27 0.11595018
## 2 1e+00 0.5 0.13 0.08232726
## 3 1e+01 0.5 0.15 0.07071068
## 4 1e+02 0.5 0.17 0.08232726
## 5 1e+03 0.5 0.21 0.09944289
## 6 1e-01 1.0 0.25 0.13540064
## 7 1e+00 1.0 0.13 0.08232726
## 8 1e+01 1.0 0.16 0.06992059
## 9 1e+02 1.0 0.20 0.09428090
## 10 1e+03 1.0 0.20 0.08164966
## 11 1e-01 2.0 0.25 0.12692955
## 12 1e+00 2.0 0.12 0.09189366
## 13 1e+01 2.0 0.17 0.09486833
## 14 1e+02 2.0 0.19 0.09944289
## 15 1e+03 2.0 0.20 0.09428090
## 16 1e-01 3.0 0.27 0.11595018
## 17 1e+00 3.0 0.13 0.09486833
## 18 1e+01 3.0 0.18 0.10327956
## 19 1e+02 3.0 0.21 0.08755950
## 20 1e+03 3.0 0.22 0.10327956
## 21 1e-01 4.0 0.27 0.11595018
## 22 1e+00 4.0 0.15 0.10801234
## 23 1e+01 4.0 0.18 0.11352924
## 24 1e+02 4.0 0.21 0.08755950
## 25 1e+03 4.0 0.24 0.10749677table(true=dat[-train,"y"], pred=predict(tune.out$best.model,newdata=dat[-train,]))## pred
## true 1 2
## 1 74 3
## 2 7 16ROC Curves
library(ROCR)## Loading required package: gplots
##
## Attaching package: 'gplots'
##
## The following object is masked from 'package:stats':
##
## lowessrocplot=function(pred, truth, ...){
predob = prediction(pred, truth)
perf = performance(predob, "tpr", "fpr")
plot(perf,...)}
svmfit.opt=svm(y~., data=dat[train,], kernel="radial",gamma=2, cost=1,decision.values=T)
fitted=attributes(predict(svmfit.opt,dat[train,],decision.values=TRUE))$decision.values
par(mfrow=c(1,2))
rocplot(fitted,dat[train,"y"],main="Training Data")
svmfit.flex=svm(y~., data=dat[train,], kernel="radial",gamma=50, cost=1, decision.values=T)
fitted=attributes(predict(svmfit.flex,dat[train,],decision.values=T))$decision.values
rocplot(fitted,dat[train,"y"],add=T,col="red")
fitted=attributes(predict(svmfit.opt,dat[-train,],decision.values=T))$decision.values
rocplot(fitted,dat[-train,"y"],main="Test Data")
fitted=attributes(predict(svmfit.flex,dat[-train,],decision.values=T))$decision.values
rocplot(fitted,dat[-train,"y"],add=T,col="red")
SVM with Multiple Classes
set.seed(1)
x=rbind(x, matrix(rnorm(50*2), ncol=2))
y=c(y, rep(0,50))
x[y==0,2]=x[y==0,2]+2
dat=data.frame(x=x, y=as.factor(y))
par(mfrow=c(1,1))
plot(x,col=(y+1))
svmfit=svm(y~., data=dat, kernel="radial", cost=10, gamma=1)
plot(svmfit, dat)
Application to Gene Expression Data
names(Khan)## [1] "xtrain" "xtest" "ytrain" "ytest"dim(Khan$xtrain)## [1] 63 2308dim(Khan$xtest)## [1] 20 2308length(Khan$ytrain)## [1] 63length(Khan$ytest)## [1] 20table(Khan$ytrain)##
## 1 2 3 4
## 8 23 12 20table(Khan$ytest)##
## 1 2 3 4
## 3 6 6 5dat=data.frame(x=Khan$xtrain, y=as.factor(Khan$ytrain))
out=svm(y~., data=dat, kernel="linear",cost=10)
summary(out)##
## Call:
## svm(formula = y ~ ., data = dat, kernel = "linear", cost = 10)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 10
## gamma: 0.0004332756
##
## Number of Support Vectors: 58
##
## ( 20 20 11 7 )
##
##
## Number of Classes: 4
##
## Levels:
## 1 2 3 4table(out$fitted, dat$y)##
## 1 2 3 4
## 1 8 0 0 0
## 2 0 23 0 0
## 3 0 0 12 0
## 4 0 0 0 20dat.te=data.frame(x=Khan$xtest, y=as.factor(Khan$ytest))
pred.te=predict(out, newdata=dat.te)
table(pred.te, dat.te$y)##
## pred.te 1 2 3 4
## 1 3 0 0 0
## 2 0 6 2 0
## 3 0 0 4 0
## 4 0 0 0 5Chapter 10
Principal Components Analysis
states=row.names(USArrests)
states## [1] "Alabama" "Alaska" "Arizona" "Arkansas"
## [5] "California" "Colorado" "Connecticut" "Delaware"
## [9] "Florida" "Georgia" "Hawaii" "Idaho"
## [13] "Illinois" "Indiana" "Iowa" "Kansas"
## [17] "Kentucky" "Louisiana" "Maine" "Maryland"
## [21] "Massachusetts" "Michigan" "Minnesota" "Mississippi"
## [25] "Missouri" "Montana" "Nebraska" "Nevada"
## [29] "New Hampshire" "New Jersey" "New Mexico" "New York"
## [33] "North Carolina" "North Dakota" "Ohio" "Oklahoma"
## [37] "Oregon" "Pennsylvania" "Rhode Island" "South Carolina"
## [41] "South Dakota" "Tennessee" "Texas" "Utah"
## [45] "Vermont" "Virginia" "Washington" "West Virginia"
## [49] "Wisconsin" "Wyoming"names(USArrests)## [1] "Murder" "Assault" "UrbanPop" "Rape"apply(USArrests, 2, mean)## Murder Assault UrbanPop Rape
## 7.788 170.760 65.540 21.232apply(USArrests, 2, var)## Murder Assault UrbanPop Rape
## 18.97047 6945.16571 209.51878 87.72916pr.out=prcomp(USArrests, scale=TRUE)
names(pr.out)## [1] "sdev" "rotation" "center" "scale" "x"pr.out$center## Murder Assault UrbanPop Rape
## 7.788 170.760 65.540 21.232pr.out$scale## Murder Assault UrbanPop Rape
## 4.355510 83.337661 14.474763 9.366385pr.out$rotation## PC1 PC2 PC3 PC4
## Murder -0.5358995 0.4181809 -0.3412327 0.64922780
## Assault -0.5831836 0.1879856 -0.2681484 -0.74340748
## UrbanPop -0.2781909 -0.8728062 -0.3780158 0.13387773
## Rape -0.5434321 -0.1673186 0.8177779 0.08902432dim(pr.out$x)## [1] 50 4biplot(pr.out, scale=0)
pr.out$rotation=-pr.out$rotation
pr.out$x=-pr.out$x
biplot(pr.out, scale=0)
pr.out$sdev## [1] 1.5748783 0.9948694 0.5971291 0.4164494pr.var=pr.out$sdev^2
pr.var## [1] 2.4802416 0.9897652 0.3565632 0.1734301pve=pr.var/sum(pr.var)
pve## [1] 0.62006039 0.24744129 0.08914080 0.04335752plot(pve, xlab="Principal Component", ylab="Proportion of Variance Explained", ylim=c(0,1),type='b')
plot(cumsum(pve), xlab="Principal Component", ylab="Cumulative Proportion of Variance Explained", ylim=c(0,1),type='b')
a=c(1,2,8,-3)
cumsum(a)## [1] 1 3 11 8Clustering: K-Means Clustering
set.seed(2)
x=matrix(rnorm(50*2), ncol=2)
x[1:25,1]=x[1:25,1]+3
x[1:25,2]=x[1:25,2]-4
km.out=kmeans(x,2,nstart=20)
km.out$cluster## [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1plot(x, col=(km.out$cluster+1), main="K-Means Clustering Results with K=2", xlab="", ylab="", pch=20, cex=2)
set.seed(4)
km.out=kmeans(x,3,nstart=20)
km.out## K-means clustering with 3 clusters of sizes 10, 23, 17
##
## Cluster means:
## [,1] [,2]
## 1 2.3001545 -2.69622023
## 2 -0.3820397 -0.08740753
## 3 3.7789567 -4.56200798
##
## Clustering vector:
## [1] 3 1 3 1 3 3 3 1 3 1 3 1 3 1 3 1 3 3 3 3 3 1 3 3 3 2 2 2 2 2 2 2 2 2 2
## [36] 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2
##
## Within cluster sum of squares by cluster:
## [1] 19.56137 52.67700 25.74089
## (between_SS / total_SS = 79.3 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"plot(x, col=(km.out$cluster+1), main="K-Means Clustering Results with K=3", xlab="", ylab="", pch=20, cex=2)
set.seed(3)
km.out=kmeans(x,3,nstart=1)
km.out$tot.withinss## [1] 104.3319km.out=kmeans(x,3,nstart=20)
km.out$tot.withinss## [1] 97.97927Clustering: Hierarchical Clustering
hc.complete=hclust(dist(x), method="complete")
hc.average=hclust(dist(x), method="average")
hc.single=hclust(dist(x), method="single")
par(mfrow=c(1,3))
plot(hc.complete,main="Complete Linkage", xlab="", sub="", cex=.9)
plot(hc.average, main="Average Linkage", xlab="", sub="", cex=.9)
plot(hc.single, main="Single Linkage", xlab="", sub="", cex=.9)
cutree(hc.complete, 2)## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
## [36] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2cutree(hc.average, 2)## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 2
## [36] 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2cutree(hc.single, 2)## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1cutree(hc.single, 4)## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3
## [36] 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3xsc=scale(x)
plot(hclust(dist(xsc), method="complete"), main="Hierarchical Clustering with Scaled Features")
x=matrix(rnorm(30*3), ncol=3)
dd=as.dist(1-cor(t(x)))
plot(hclust(dd, method="complete"), main="Complete Linkage with Correlation-Based Distance", xlab="", sub="")
NCI60 Data Example: The NCI60 data
nci.labs=NCI60$labs
nci.data=NCI60$data
dim(nci.data)## [1] 64 6830nci.labs[1:4]## [1] "CNS" "CNS" "CNS" "RENAL"table(nci.labs)## nci.labs
## BREAST CNS COLON K562A-repro K562B-repro LEUKEMIA
## 7 5 7 1 1 6
## MCF7A-repro MCF7D-repro MELANOMA NSCLC OVARIAN PROSTATE
## 1 1 8 9 6 2
## RENAL UNKNOWN
## 9 1NCI60 Data Example: PCA on the NCI60 Data
pr.out=prcomp(nci.data, scale=TRUE)
Cols=function(vec){
cols=rainbow(length(unique(vec)))
return(cols[as.numeric(as.factor(vec))])
}
par(mfrow=c(1,2))
plot(pr.out$x[,1:2], col=Cols(nci.labs), pch=19,xlab="Z1",ylab="Z2")
plot(pr.out$x[,c(1,3)], col=Cols(nci.labs), pch=19,xlab="Z1",ylab="Z3")
summary(pr.out)## Importance of components:
## PC1 PC2 PC3 PC4 PC5
## Standard deviation 27.8535 21.48136 19.82046 17.03256 15.97181
## Proportion of Variance 0.1136 0.06756 0.05752 0.04248 0.03735
## Cumulative Proportion 0.1136 0.18115 0.23867 0.28115 0.31850
## PC6 PC7 PC8 PC9 PC10
## Standard deviation 15.72108 14.47145 13.54427 13.14400 12.73860
## Proportion of Variance 0.03619 0.03066 0.02686 0.02529 0.02376
## Cumulative Proportion 0.35468 0.38534 0.41220 0.43750 0.46126
## PC11 PC12 PC13 PC14 PC15
## Standard deviation 12.68672 12.15769 11.83019 11.62554 11.43779
## Proportion of Variance 0.02357 0.02164 0.02049 0.01979 0.01915
## Cumulative Proportion 0.48482 0.50646 0.52695 0.54674 0.56590
## PC16 PC17 PC18 PC19 PC20
## Standard deviation 11.00051 10.65666 10.48880 10.43518 10.3219
## Proportion of Variance 0.01772 0.01663 0.01611 0.01594 0.0156
## Cumulative Proportion 0.58361 0.60024 0.61635 0.63229 0.6479
## PC21 PC22 PC23 PC24 PC25 PC26
## Standard deviation 10.14608 10.0544 9.90265 9.64766 9.50764 9.33253
## Proportion of Variance 0.01507 0.0148 0.01436 0.01363 0.01324 0.01275
## Cumulative Proportion 0.66296 0.6778 0.69212 0.70575 0.71899 0.73174
## PC27 PC28 PC29 PC30 PC31 PC32
## Standard deviation 9.27320 9.0900 8.98117 8.75003 8.59962 8.44738
## Proportion of Variance 0.01259 0.0121 0.01181 0.01121 0.01083 0.01045
## Cumulative Proportion 0.74433 0.7564 0.76824 0.77945 0.79027 0.80072
## PC33 PC34 PC35 PC36 PC37 PC38
## Standard deviation 8.37305 8.21579 8.15731 7.97465 7.90446 7.82127
## Proportion of Variance 0.01026 0.00988 0.00974 0.00931 0.00915 0.00896
## Cumulative Proportion 0.81099 0.82087 0.83061 0.83992 0.84907 0.85803
## PC39 PC40 PC41 PC42 PC43 PC44
## Standard deviation 7.72156 7.58603 7.45619 7.3444 7.10449 7.0131
## Proportion of Variance 0.00873 0.00843 0.00814 0.0079 0.00739 0.0072
## Cumulative Proportion 0.86676 0.87518 0.88332 0.8912 0.89861 0.9058
## PC45 PC46 PC47 PC48 PC49 PC50
## Standard deviation 6.95839 6.8663 6.80744 6.64763 6.61607 6.40793
## Proportion of Variance 0.00709 0.0069 0.00678 0.00647 0.00641 0.00601
## Cumulative Proportion 0.91290 0.9198 0.92659 0.93306 0.93947 0.94548
## PC51 PC52 PC53 PC54 PC55 PC56
## Standard deviation 6.21984 6.20326 6.06706 5.91805 5.91233 5.73539
## Proportion of Variance 0.00566 0.00563 0.00539 0.00513 0.00512 0.00482
## Cumulative Proportion 0.95114 0.95678 0.96216 0.96729 0.97241 0.97723
## PC57 PC58 PC59 PC60 PC61 PC62
## Standard deviation 5.47261 5.2921 5.02117 4.68398 4.17567 4.08212
## Proportion of Variance 0.00438 0.0041 0.00369 0.00321 0.00255 0.00244
## Cumulative Proportion 0.98161 0.9857 0.98940 0.99262 0.99517 0.99761
## PC63 PC64
## Standard deviation 4.04124 2.148e-14
## Proportion of Variance 0.00239 0.000e+00
## Cumulative Proportion 1.00000 1.000e+00plot(pr.out)
pve=100*pr.out$sdev^2/sum(pr.out$sdev^2)
par(mfrow=c(1,2))
plot(pve, type="o", ylab="PVE", xlab="Principal Component", col="blue")
plot(cumsum(pve), type="o", ylab="Cumulative PVE", xlab="Principal Component", col="brown3")
NCI60 Data Example: Clustering the Observations of the NCI60 Data
sd.data=scale(nci.data)
par(mfrow=c(1,3))
data.dist=dist(sd.data)
plot(hclust(data.dist), labels=nci.labs, main="Complete Linkage", xlab="", sub="",ylab="")
plot(hclust(data.dist, method="average"), labels=nci.labs, main="Average Linkage", xlab="", sub="",ylab="")
plot(hclust(data.dist, method="single"), labels=nci.labs, main="Single Linkage", xlab="", sub="",ylab="")
hc.out=hclust(dist(sd.data))
hc.clusters=cutree(hc.out,4)
table(hc.clusters,nci.labs)## nci.labs
## hc.clusters BREAST CNS COLON K562A-repro K562B-repro LEUKEMIA MCF7A-repro
## 1 2 3 2 0 0 0 0
## 2 3 2 0 0 0 0 0
## 3 0 0 0 1 1 6 0
## 4 2 0 5 0 0 0 1
## nci.labs
## hc.clusters MCF7D-repro MELANOMA NSCLC OVARIAN PROSTATE RENAL UNKNOWN
## 1 0 8 8 6 2 8 1
## 2 0 0 1 0 0 1 0
## 3 0 0 0 0 0 0 0
## 4 1 0 0 0 0 0 0par(mfrow=c(1,1))
plot(hc.out, labels=nci.labs)
abline(h=139, col="red")
hc.out##
## Call:
## hclust(d = dist(sd.data))
##
## Cluster method : complete
## Distance : euclidean
## Number of objects: 64set.seed(2)
km.out=kmeans(sd.data, 4, nstart=20)
km.clusters=km.out$cluster
table(km.clusters,hc.clusters)## hc.clusters
## km.clusters 1 2 3 4
## 1 11 0 0 9
## 2 0 0 8 0
## 3 9 0 0 0
## 4 20 7 0 0hc.out=hclust(dist(pr.out$x[,1:5]))
plot(hc.out, labels=nci.labs, main="Hier. Clust. on First Five Score Vectors")
table(cutree(hc.out,4), nci.labs)## nci.labs
## BREAST CNS COLON K562A-repro K562B-repro LEUKEMIA MCF7A-repro
## 1 0 2 7 0 0 2 0
## 2 5 3 0 0 0 0 0
## 3 0 0 0 1 1 4 0
## 4 2 0 0 0 0 0 1
## nci.labs
## MCF7D-repro MELANOMA NSCLC OVARIAN PROSTATE RENAL UNKNOWN
## 1 0 1 8 5 2 7 0
## 2 0 7 1 1 0 2 1
## 3 0 0 0 0 0 0 0
## 4 1 0 0 0 0 0 0EOF